SQUARE-ROOT CUBATURE-QUADRATURE KALMAN FILTER

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1 Asian Journal of Control, Vol. 6, No. 2, pp , March 204 Published online 8 April 203 in Wiley Online Library (wileyonlinelibrary.com) DOI: 0.002/asjc.704 SQUARE-ROO CUBAURE-QUADRAURE KALMAN FILER Shovan Bhaumi and Swati Brief Paper ABSRAC In this paper, an on-going wor introducing square-root extension of cubature-quadrature based Kalman filter is reported. he proposed method is named square-root cubature-quadrature Kalman filter (SR-CQKF). Unlie ordinary cubature-quadrature Kalman filter (CQKF), the proposed method propagates and updates square-root of the error covariance without performing Cholesy decomposition at each step. Moreover SR-CQKF ensures positive semi-definiteness of the state covariance matrix. With the help of two examples we show the superior performance of SR-CQKF compared to and square root cubature Kalman filter. Key Words: Kalman filter, Gauss quadrature rule, square-root filter, nonlinear estimation. I. INRODUCION Many researchers have devoted their best efforts to develop computationally affordable efficient algorithm for nonlinear estimation because it is essential in many real-life problems. Until recently, the extended Kalman filter () has been the natural choice of practitioners for solving nonlinear problems. he uses first order linearization to approximately calculate the mean and covariance of non Gaussian probability density function. Due to such crude approximation, the filter looses trac if the nonlinearity or uncertainty in the system is high. Limitations of for severely nonlinear problems are well documented in earlier literature [,2]. o alleviate the problem, in [3] we proposed a technique named cubature-quadrature Kalman filter (CQKF). CQKF is based on spherical cubature and Gauss-Laguerre quadrature rule of numerical integration. he proposed technique is a more generalized form of cubature Kalman filter (CKF) [4], (which we here call CQKF-) and under single Gauss-Laguerre quadrature point approximation it merges with CKF. In CQKF, it is necessary to perform Cholesy decomposition at each step. Due to accumulated round off error associated with processing software, Cholesy decomposition sometimes leads to negative definite covariance matrices, causing the filter to stop. Unscented Kalman filter (UKF) [5] and Gauss Hermite filter (GHF) also suffer from similar type of hitch which has been rectified by Merwe et al. [6], and Arasaratnam et al. [7], by proposing square-root formulation. In [7], Arasaratnam and Heyin proposed a square root version of the Gauss-Hermite filter algorithm which uses the Gauss-Hermite quadrature method to evaluate intractable integrals. In the present paper, we propose the square root version of cubature quadrature filter which uses third order cubature and Gauss-Laguerre quadrature Manuscript received June 3, 202; revised October 5, 202; accepted November 4, 202. he authors are with Department of Electrical Engineering, Indian Institute of echnology Patna, Bihar-80003, India Dr Shovan Bhaumi is the corresponding author ( iitp.ac.in). rule to evaluate the integrals. he methodology adopted here to estimate the states of a nonlinear system is totally different from [7], whereas the square root method we formulate is similar to [6] and [7]. he square-root version of UKF is now being applied to practical problems [8,9] for enhanced computational stability. Motivated by earlier wors [6,7] and their usefulness, we propose the square-root version of CQKF (SR-CQKF) which enhances numerical stability by ensuring positive semi-definiteness of the error covariance matrix. Unlie the CQKF, where Cholesy decomposition, computationally the most expensive operation, has to be performed at each step, the SR-CQKF propagates the state estimate and its square-root covariance. With the help of the examples, we show that SR-CQKF outperforms and. he rest of the paper is organized as follows. he next section provides the brief description of CQKF algorithm. his is followed by elaborate algorithm of SR-CQKF. Finally simulation results, and concluding remars are given in Section 4 and 5 respectively. II. CUBAURE-QUADRAURE KALMAN FILER We developed a novel algorithm [3] where the multivariate moment integral has been computed numerically using cubature rule and multiple Gauss-Laguerre quadrature points. he algorithm has been named the cubature-quadrature Kalman filter. In this section, we briefly describe the algorithm. For detailed derivation readers are referred to [3]. Let us consider a nonlinear plant described by the state and measurement equations as follows: x+ = φ( x) + η () y = γ ( x ) + v (2) where x R n denotes the state of the system, y R p is the measurement at the instant where = {0,, 2, 3,... N},f(x ) 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

2 68 Asian Journal of Control, Vol. 6, No. 2, pp , March 204 and g (x ) are nown nonlinear functions of x and. he process noise h R n and measurement noise v R p are assumed to be uncorrelated and normally distributed with covariance Q and R respectively. CQKF approximately evaluates the intractable integrals encountered in Baysian filtering framewor using cubaturequadrature points and corresponding weights associated with them. For n dimensional problem solved with third order spherical cubature and n order of quadrature rule, total number of points and associated weights need to be calculated. he algorithm of the cubature-quadrature Kalman filter (CQKF) could be summarized as follows: Step i. Filter initialization. Initialize the filter with xˆ 00 and P 0 0. Calculate the cubature-quadrature (CQ) points x j and their corresponding weights w j( j =, 2,..., ). Step ii. Predictor step. Perform Cholesy decomposition of posterior error covariance P = S S (3) = S ξ + xˆ (4) j, j Update cubature-quadrature points φ( ) (5) j, + = j, Compute time updated mean and covariance ˆx = w (6) + j j, + + j[ j, + + ] j, + + [ ] + P = w xˆ xˆ Q (7) Step iii. Corrector step or measurement update. Perform Cholesy decomposition of prior error covariance P = S S (8) = S ξ + xˆ (9) j, + + j + where j =,2,.... Find the predicted measurements at each cubaturequadrature points Y j, + = j, + γ( ) (0) Estimate the predicted measurement ŷ = w Y, () + j j + Calculate the covariances [ ] + Py y = wj Yj y Yj y R + +, + ˆ +, + ˆ + (2) [ ] Px y = wj j x Yj y + +, + ˆ +, + ˆ + (3) Calculate Kalman gain K P P [ ][ ] + = x+ y+ y+ y (4) + Compute posterior state values xˆ+ + = xˆ+ + K+ ( y+ yˆ+ ) (5) Posterior error covariance matrix is given by P+ + = P+ K+ Py y K (6) III. SQUARE-ROO CUBAURE-QUADRAURE KALMAN FILER he SR-CQKF is an algebraic re-formulation of CQKF algorithm, where Cholesy decomposition need not to be performed at each step. Due to limited word length in processing software, round off error accumulates and after some iterations the positive semi-definite nature of the error covariance matrix may be lost. o circumvent the problem, in this section, we re-formulate the CQKF algorithm based on orthogonal-triangular decomposition, popularly nown as QR decomposition. he error covariance matrix, P can be factorized as, P = AA (7) where P R n n, and A R n m with m n. Due to increased dimension of A, Arasaratnam and Hayin [7] called it as fat matrix. he QR decomposition of matrix A is given by, A = QR where Q R m m is orthogonal, and R R m n is upper triangular. With the QR decomposition described above, the error covariance matrix can be written as P = AA = R Q QR= R R = SS (8) R is the upper triangular part of R matrix, n n R R, where R = S. he detailed algorithm of square-root cubature-quadrature Kalman filter is provided below. We use the notation qr{} to denote the QR decomposition of a matrix and uptri{} to select the upper triangular part of a matrix. 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

3 S. Bhaumi and Swati: Square-Root Cubature-Quadrature Kalman Filter 69 Step i. Filter initialization. Initialize the filter with xˆ 00 and P00 = S00 S00 Calculate the cubature-quadrature (CQ) points x j and their corresponding weights w j (j =, 2,..., ). he methodology to calculate n order cubature quadrature points and associated weights has been described in paper [3]. Step ii. Predictor step. = S ξ + xˆ (9) j, j Update cubature-quadrature points φ( ) (20) j, + = j, Compute time updated mean ˆx = w (2) + j j, + Evaluate square-root weight matrix W = w 0 0 w2 nn (22) Calculate the matrix of square-root weighted cubature-quadrature points after prior mean subtraction [ ] * = xˆ xˆ xˆ W +, + + 2, + +, + + Perform QR decomposition { } (23) R = + qr * + Q (24) Estimate the square-root of predicted error covariance S = uptri{ R } (25) + + Step iii. Measurement update. = S ξ + xˆ (26) j, + + j + Find the predicted measurements at each CQ points Y j, + = j, + γ( ) (27) Estimate the predicted measurement ŷ = w Y, (28) + j j + Calculate the square-root weighted measurement matrix [ ] Y* = Y yˆ Y yˆ Y yˆ W +, + + 2, + +, + + Perform QR decomposition { } (29) R = y + y qr Y * + + R (30) Calculate the square root of innovation covariance S = uptri{ R } (3) y+ y+ y+ y+ Calculate the cross covariance matrix P * Y* (32) x y = Calculate Kalman gain K P S S = ( ) x y y y y y (33) Compute posterior state values xˆ+ + = xˆ+ + K( y+ yˆ+ ) (34) Calculate QR decomposition { } R = + + qr * + KY * + Q K R (35) Calculate square root of posterior error covariance matrix S = uptri{ R } (36) Note. he equation (35) can be derived from equations (4), (6), (24), (30), and (32) by matrix manipulation. For detailed steps readers are refered to [7]. Note 2. he practitioners who want to implement the algorithm in MALAB software environment, should note that the command chol (P) returns the matrix S (where P = SS ) rather than S. Accordingly the expressions for next steps need to be evaluated. IV. SIMULAION RESULS In this section, the formulated algorithm described above has been applied to solve a single dimensional as well as multi dimensional nonlinear problem. Example. Single dimensional process. he plant model, inspired by [0], has one unstable and two stable equilibrium points and given by x+ = φ( x) + η 2 2 φ( x) = x+ Δt5x( x ), η ~ ℵ( 0, b Δt) 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

4 620 Asian Journal of Control, Vol. 6, No. 2, pp , March 204 he measurement model is y = γ ( x ) + v 2 γ ( x) = Δtx( 0. 5x), v ~ ℵ( 0, d Δt) and the value of Dt = 0.0 seconds, x 0 =-0.2, xˆ 00 = 08., P 0 0 = 2, b = 0.5 and d = 0.. We have considered the time span from 0 to 4 seconds. he system has three equilibrium points, of which, the one at the origin is unstable and the other two are at and stable. In the absence of any bias, the system settles around either of the two stable equilibrium points. he problem becomes challenging because moderate estimation error forces the estimate to settle down at the wrong equilibrium point, leading to a trac loss situation. We solve the estimation problem using and SR-CQKF. he percentage fail counts for different filters are shown in able I. he percentage fail count is defined as the number of cases where estimation error at 4th second is more than unity out of 00 Monte Carlo runs. In our notation, CQKF-x indicates an x order Gauss-Laguerre cubature-quadrature filter. Under the first order Gauss-Laguerre approximation CQKF reduces to CKF. he numbers mentioned in the table show the improvement of performance with the square-root CQKF. able I. Percentage fail count. Filter Fail Count 22% or SR-CQKF- 4%.05%.02%.02% Fig. shows the root mean square error () obtained from SR-CQKF and out of 0,000 Monte Carlo runs. It can be seen from Fig., that of higher order SR-CQKF converges more quicly than first order SR-CQKF or. However no significant improvement of performance is noticed with more than second order SR-CQKF. Example 2. Lorenz system. Inspired from earlier wors [0,], in this example, we consider the discrete time Lorenz attractor, one of the classic icons in nonlinear dynamics. he attractor is available in the literature from 963, mainly due to the wor of meteorologist Edward Lorenz. Although a higher dimensional Lorenz attractor may exists [2,3] in real life problems, here we consider three dimensional system. he process equation is given by x+ = φ( x) + bη φ( x) = x+ Δtf( x), η ~ ℵ( 0, Δt) he measurement equation is expressed as y = γ ( x ) + dv γ ( x) = Δth( x), v ~ ℵ( 0, Δt) Here x = [x, x 2, x 3,] R 3 is state variable. he parameters b R 3, and d R are constant whose values are taen as b = [ ] and d = he functions f(x) and h(x) are given by [ ] f( x) = α( x + x ) βx x x x γx + x x ime (s) Fig.. plot of SR-CQKF with higher order Gauss-Laguerre quadrature points. 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

5 S. Bhaumi and Swati: Square-Root Cubature-Quadrature Kalman Filter 62 able II. Average root mean square error over time span. Filter Averaged Averaged Averaged (State ) (State 2) (State 3) or SR-CQKF and ime (sec) Fig. 2. plot of and SR-CQKF for first state variable. h( x)= x 2 + x2 2 + x3 2 he three parameters, a (called the Prandtl number), b (called the Rayleigh number), and g have great impact on the system. he system has three unstable equilibrium points and for a 0, and g(b - ) 0 they are at [0 0 0], γ( β ) γ( β ) ( β ), and γ( β ) γ( β ) ( β ). We chose the system with classical parameter values: a = 0, g = 8/3, and b = 28, for which almost all points in phase space tend to a strange attractor [3]. he initial truth value of state is taen as x 0 = [ ]. We simulate during the time interval of 0 to 4 seconds with the sampling time Dt = 0.0 seconds. he initial posterior estimate is assumed as xˆ 00 = [ ] with initial error covariance P 0 0 = 0.35I 3. he above described problem has been solved using square-root version of CQKF with higher order of Gauss-Laguerre approximation. We observe that sometimes ordinary CQKF fails to run due to non positive definite error covariance matrix. he same problem does not occur during SR-CQKF as expected from the formulation. he root mean square error, averaged over time span, obtained from 50 Monte Carlo runs, has been tabulated in able II. In Figs 2 4, we show the root mean square errors (s) of the states obtained from and SR-CQKF. was calculated from 50 Monte Carlo ime (sec) Fig. 3. plot of and SR-CQKF for second state variable ime (sec) Fig. 4. plot of and SR-CQKF for third state variable. runs. he results obtained from the simulation run may be summarized as follows: From Fig. 4, no improvement in estimation of third state variable has been observed with SR-CQKF in comparison with. his is also supported from the numbers displayed in able II. uses first order linearization to approximately calculate the mean and covariance of non Gaussian probability density function. Cubature quadrature Kalman filter uses cubature rule and Gauss-Laguerre quadrature points to 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

6 622 Asian Journal of Control, Vol. 6, No. 2, pp , March 204 determine the mean and covariance of posterior probability density function. he moment calculation is more accurate in the second case; hence improvement in performance is expected compared to. he argument is well supported by Figs 2 3, where we notice that for first and second state variable, SR-CQKF- or performs significantly better than. Further for the first and second state variable, performs considerably better than SR-CQKF-. Although from the Figs 2 3 no noticeable improvement is observed with SR-CQKF with the order higher than 2. From able II, it is clear that SR-CQKF provides significantly lower averaged in comparison to. Also the performance of the SR-CQKF improves with the higher order Gauss-Laguerre quadrature approximation. However after the 4th order, SR-CQKF does not offer any improvement, hence it is not displayed in the table. Computational times of SR-CQKF and CQKF have been compared. It has been observed that CQKF- and SR-CQKF- tae 7.83 seconds and 4.9 seconds respectively for 50 MC runs in MALAB software in a personal computer with a 2.40 GHz processor. he result indicates lower computational cost of square root cubature quadrature Kalman filter compared to ordinary CQKF. V. DISCUSSIONS AND CONCLUSION In this paper, the square-root extension of cubaturequadrature Kalman filter has been proposed. In comparison to CQKF, the proposed estimator has better numerical properties and guaranteed positive semi-definiteness of error covariance matrix. Instead of performing Cholesy decomposition at each step, the proposed filter propagates and updates square-root of the error covariance. With the help of examples, the superiority of the proposed algorithm over and is demonstrated. he simulation results also reveal the performance improvement of SR-CQKF with higher order Gauss-Laguerre quadrature approximation. REFERENCES. Julier, S. J. and J. K. Uhlmann, Unscented filtering and nonlinear estimation, Proceedings of the IEEE, Vol. 92, No. 3, (2004). 2. Luca, M., S. Azou, G. Burel, and A. Serbanescu, On exact Kalman filtering of polynomial systems, IEEE rans. Circuits Syst. Regul. Pap., Vol. 53, No. 6, pp (2006). 3. Swati and S. Bhaumi, Non linear estimation using cubature quadrature points, Proc. IEEE International Conference on Energy, Automation and Signal, Bhubaneswar, India, Dec (20). 4. Arasaratnam, I. and S. Hayin, Cubature Kalman filters, IEEE rans. Autom. Control, Vol. 54, No. 6, (2009). 5. Julier, S., J. Uhlmann, and D. Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE rans. Autom. Control, Vol. 45, No. 3, pp (2000). 6. Van der Merwe, R. and E. A. Wan, he square-root unscented Kalman filter for state and parameter-estimation, Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol. 6, pp (200). 7. Arasaratnam, I. and S. Hayin, Square-root quadrature Kalman filtering, IEEE rans. Signal Process., Vol. 56, No. 6, pp (2008). 8. ang, Xiaojun, Jie Yan, and Dudu Zhong. Square-root sigma-point Kalman filtering for spacecraft relative navigation, Acta Astronaut., Vol 66, pp (200). 9. Soen, H. E. and C. Hajiyev, UKF based In-flight calibration of magnetometers and rate gyros for pico satellite attitude determination, Asian J. Control, Vol. 4, No. 3, pp (202). 0. Ito, K. and K. Xiong, Gaussian filters for nonlinear filtering problems, IEEE rans. Autom. Control, Vol. 45, No. 5, pp (2000).. erejanu, G., P. Singla,. Singh, and P. D. Scott, Adaptive gaussian sum filter for nonlinear bayesian estimation, IEEE rans. Autom. Control, Vol. 56, No.9, pp (20). 2. Stewart, I., he lorenz attractor exists, Nature, Vol. 406, pp (2000). 3. ucer, W. he lorentz attractor exists, C.R. Acad. Sci. Paris Ser. I Math., Vol. 328, No. 2, pp (999). 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

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